UBC Mathematics Department

Colloquium Abstract: Professor Walter Rudin, Department of Mathematics, University of Wisconsin-Madison

Analytic aspects of the Bohr Topology

If G is an abelian group, then G^# denotes G equipped with the weakest topology that makes every character of G continuous. This is the Bohr topology of G. If G= \Bbb Z, the additive group of the integers, and A is a Hadamard set in \Bbb Z, it is shown that: (i) A-A has 0 as its only limit point in \Bbb Z^#, (ii) No Sidon subset of A-A has a limit point in \Bbb Z^#, (iii) A-A is a \Lambda (p) set for all p<\infty. This leads to an explicit example of a set which is \Lambda (p) for all p<\infty and is dense in \Bbb Z^#. If f(x) is a quadratic or cubic polynomial with integer coefficients, then the closure of f(\Bbb Z) in the Bohr compactification of \Bbb Z is shown to have Haar measure 0.

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